Homework Help:
Simple problem - contracting tensors

1. The problem statement, all variables and given/known data
Hello guys, hope you'll help me out with this!!
I'm asked to calculate [tex]g^{\alpha\beta}g^{\sigma\rho}(g_{\alpha\sigma}g_{\beta\rho}-g_{\alpha\rho}g_{\beta\sigma})[/tex]
where g is the metric tensor on a n-dimensional manifold but I can't get to the right result, i keep on getting zero!! (i know that's wrong although i dont know the exact solution -sorry- but it should depend on the dimension of the manifold)

2. Relevant equations

none

3. The attempt at a solution

Well, i contracted firs with respect to [itex]g^{\sigma\rho}[/itex] and i end up with [tex]g^{\alpha\beta}(g_{alpha\beta}+g_{alpha\beta}-g_{alpha\beta}-g_{alpha\beta})[/tex]which obviously vanish.
I must be missing something!!!

Are you so sure it's zero? Let's look at the first factor:
[tex]
g^{\alpha\beta}g^{\sigma\rho}g_{\alpha\sigma}g_{ \beta\rho}=g^{\alpha\beta}(g^{\sigma\rho}g_{\alpha\sigma})g_{\beta\rho}=g^{\alpha\beta}\delta^{\rho}_{\alpha}g_{\beta\rho}=g^{\rho\beta}g_{\beta\rho}=\delta^{\rho}_{\rho}=n
[/tex]
What is the other factor?

That's becaouse contracting the second factor like you did for the first i get
[tex]-g^{\alpha\beta} g^{\sigma\rho}g_{\alpha\rho}g_{\beta\sigma}=-g^{\alpha\beta} (g^{\sigma\rho}g_{\alpha\rho})g_{\beta\sigma}=-g^{\alpha\beta} \delta^{\sigma}_{\alpha}g_{\beta\sigma}=-g^{\sigma\beta}g_{\beta\sigma}=-n[/tex]

which added to the first factor gives zero!
They seems equal to me, becaouse when i get the kronecker delta it doesent matter which index it has, it will be a mute index anyhow! (maybe the error is here?)